Week 4: Preparation#

Reading Material#

We recommend that you read the textbook. Watching YouTube videos on the week’s topics can be useful, but it should not replace proper preparation for the week’s program and is not recommended as a standalone approach.

Read and study the following:

Key Concepts#

After reading, you should be able to explain the following key concepts:

This week, we will explore these key concepts in great detail. We expect you to have familiarized yourself with these topics before lectures.


Preparatory Exercises#

I: Identifying Type of Matrix#

Consider a \(2\times 2\) matrix given by:

\[\begin{equation*} A = \begin{bmatrix} 2 & 1-i \\ 1+i & 3 \end{bmatrix}. \end{equation*}\]
  1. Is \(A\) symmetric?

  2. Is \(A\) Hermitian?

  3. Is \(A\) normal?

II: Diagonalization of a Symmetric \(2\times 2\) Matrix#

Consider the real symmetric matrix

\[\begin{equation*} B = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}. \end{equation*}\]
  1. Find the eigenvalues of \(B\).

  2. Find for each eigenvalue an associated eigenvector.

  3. Normalize the eigenvectors.

  4. Show that \(B\) is orthogonally diagonalizable.